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Efficient simulation of tail probabilities of sums of dependent random variables

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jose H. Blanchet  and
Leonardo Rojas-Nandayapa
Jose H. Blanchet
Affiliation:
Columbia University, Department of Industrial Engineering and Operations Research, Columbia University, S.W. Mudd Building, 500 West 120th Street, New York, USA. Email address: [email protected]
Leonardo Rojas-Nandayapa
Affiliation:
The University of Queensland, Department of Mathematics, The University of Queensland, Brisbane, Queensland 4072, Australia. Email address: [email protected]
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Abstract

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We study asymptotically optimal simulation algorithms for approximating the tail probability of P(eX1+⋯+ eXd>u) as u→∞. The first algorithm proposed is based on conditional Monte Carlo and assumes that (X1,…,Xd) has an elliptical distribution with very mild assumptions on the radial component. This algorithm is applicable to a large class of models in finance, as we demonstrate with examples. In addition, we propose an importance sampling algorithm for an arbitrary dependence structure that is shown to be asymptotically optimal under mild assumptions on the marginal distributions and, basically, that we can simulate efficiently (X1,…,Xd|Xj >b) for large b. Extensions that allow us to handle portfolios of financial options are also discussed.

Keywords

Rare-event simulationefficiencydependenceheavy-tailed distributionlog-elliptical distributiontail probabilityvariance reductionimportance samplingconditional Monte Carlo

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G70: Extreme value theory; extremal processes
Type
Part 4. Simulation
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 147 - 164
Copyright
Copyright © Applied Probability Trust 2011 

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